Properties

Complex representations

In even d=2nd = 2n there are two inequivalent complex-linear irreducible representations of Spin(d−1,1)Spin(d-1,1), each of complex dimension2d/2−12^{d/2-1}, called the two chiral representations, or the two Weyl spinor representations.

For instance for d=10d = 10 one often writes these as 16\mathbf{16} and 16′\mathbf{16}'.

The direct sum of the two chiral representation is called the Dirac spinor representation, for instance 16+16′\mathbf{16} + \mathbf{16}'.

In odd d=2n+1d = 2n+1 there is a single complex irreducible representation of complex dimension2(d−1)/22^{(d-1)/2}. For instance for d=11d = 11 one often writes this as 32\mathbf{32}. This is called the Dirac spinor representation in this odd dimension.

For d=2nd = 2n, if {Γ1,⋯,Γn}\{\Gamma^1, \cdots, \Gamma^n\} denote the generators of the Clifford algebraCld−1,1Cl_{d-1,1} then there is the chirality operator

on the Dirac representation, whose eigenspaces induce its decomposition into the two chiral summands.

The unique irreducible Dirac representation in the odd dimension d+1d+1 is, as a complex vector space, the sum of the two chiral representations in dimension dd, with the Clifford algebra represented by Γ1\Gamma^1 through Γd\Gamma^d acting diagonally on the two chiral representations, and the chirality operator Γd+1\Gamma^{d+1} in dimension dd acting on their sum, now being the representation of the (d+1)(d+1)st Clifford algebra generator.

Real representations (Majorana representations)

One may ask in which dimensions dd the above complex representations admit a real structure

p=0p = 0 – spinor metric

We discuss spinor bilinear pairings to scalars.

Over the complex numbers

Proposition

Let VV be a quadratic vector space, def. 1 over the complex numbers of dimensiondd. Then there exists in dimensions d≠2,6mod8d \neq 2,6 \; mod \, 8, up to rescaling, a unique Spin(V)Spin(V)-invariant bilinear form

In such a notation if ϕ=(ϕα)\phi = (\phi^\alpha) denotes the component-vector of a spinor, then the result of “lowering its index” is given by acting with the metric in form of the charge conjugation matrix. The result is traditionally denoted

to indicate that there are N+N_+ copies of the irreducible Spin(V)Spin(V)-representation of one chirality, and N−N_- of those of the other chirality (i.e. left and right handed Weyl spinors).

This counting however is more subtle over the real numbers (Majorana spinors) and the notation in this case (which happens to be the more important case) is not entirely consistent through the literature.

There is no issue in those dimensions in which the complex Weyl representation already admits a real structure itself, hence when there are Majorana-Weyl spinors. In this case one just counts them with N+N_+ and N−N_- as in the case over the complex numbers.

However, in some dimensions it is only the direct sum of two Weyl spinor representations which carries a real structure. For instance for d=4d = 4 and d=8d = 8 in Lorentzian signature (see the above table) it is the complex representations 2⊕2′\mathbf{2} \oplus \mathbf{2}' and 16⊕16′\mathbf{16} \oplus \mathbf{16}', respectively, which carry a real structure. Hence the real representation underlying this parameterizes N=1N = 1 supersymmetry in terms Majorana spinors, even though its complexification would be N=(1,1)N = (1,1). See for instance (Freed 99, p. 53).

Similarly in dimensions 5,6 and 7 mod 8, the minimal real representation is obatained from tensoring the complex spinors with the complex 2-dimensional canonical quaternionic representation WW (as in the above table). These are also called symplectic Majorana representations. For instance in in 6d one typically speaks of the 6d (2,0)-superconformal QFT to refer to that with a single “symplectic Majorana-Weyl” supersymmetry (e.g. Figueroa-OFarrill, p. 9), which might therefore be counted as (1,0)(1,0) real supersymmetric, but which involves two complex irreps and is hence often denoted counted as (2,0)(2,0).

For 𝕂\mathbb{K} one of the four real normed division algebras, write (−)*:𝕂→≃𝕂op(-)^\ast \colon \mathbb{K} \stackrel{\simeq}{\to} \mathbb{K}^{op} for the conjugation anti-automorphism.

the real part of an element a∈𝕂a \in \mathbb{K} is Re(a)≔12(a+a*)Re(a) \coloneqq \tfrac{1}{2} (a + a^\ast);

say that an n×nn \times nmatrix with coefficients in 𝕂\mathbb{K}, A∈Matn×n(magthbbK)A\in Mat_{n\times n}(\magthbb{K}) is a hermitian matrix if the transpose matrix equals the conjugated matrix: At=A*A^t = A^\ast. Hence with (−)†≔((−)t)*(-)^\dagger \coloneqq ((-)^t)^\ast this is A=A†A = A^\dagger, as usual;

write A˜≔A−(trA)𝟙n×n\tilde A \coloneqq A - (tr A) \mathbb{1}_{n\times n} for the matrix minus its trace times the identity matrix.